# What is the derivative of a definite integral?

## What is the derivative of a definite integral?

Restating the Fundamental Theorem That is, the derivative of a definite integral of f whose upper limit is the variable x and whose lower limit is the constant a equals the function f evaluated at x. This is true regardless of the value of the lower limit a.

## When can you move a derivative inside an integral?

Observe that, in Leibniz Integral Rule if the limits are independent of the independent variable, their partial differentiation wrt that variable becomes 0, thus leaving you with the derivative inside the integral.

How do you calculate anti – derivative?

To find the anti-derivative of a particular function, find the function on the left-hand side of the table and find the corresponding antiderivative in the right-hand side of the table. For example, if the antiderivative of cos(x) is required, the table shows that the anti-derivative is sin(x) + c.

### What is the first partial derivative?

As with ordinary derivatives, a first partial derivative represents a rate of change or a slope of a tangent line. For a three-dimensional surface, two first partial derivatives represent the slope in each of two perpendicular directions. Second, third, and higher partial derivatives give more information about how the function changes at any point.

### What are partial derivatives?

Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation. (1)

What is differentiation of integral?

Differentiation Under the Integral Sign Differentiation under the integral sign is an operation in calculus used to evaluate certain integrals . Under fairly loose conditions on the function being integrated, differentiation under the integral sign allows one to interchange the order of integration and differentiation.